The Synergistic Dynamics of Art, Mathematics, & Science
An Interdisciplinary Project at Stowe Middle School
[©Frederick David Abraham, project contributors, and sources]

Answers to Exercises 1-12 on Spirals

Ex1. 360°, 720°, 1089°

Ex2. 30°, 69°, 90°, 15°, 30°, 45°

Ex3. 0, 180, 360, 540, 720, 900, . . . and so on

24,000 miles: they will pass, be tired, and have paddled and skied the same amount, if they started from exactly the same place anywhere on earth.

Ex4. 37

Ex5. 90, 180, 270

Ex6. Some of the countries are:

• 0°: United Kingdom, France, Spain, Algeirs, Mali, Bourkina Fasso, Ghana


• 90$#176: Russia, Mongolia, China, India, Bangladesh


• 180°: Russia, Kiribati, Tuvalu, Fiji


• 270° = 90°W: Canada, United States, Mexico, Guatamala, El Salvador

Ex7. Archimedean. These are points on the one turn in 360° case, and the increase in distance is the same for equal changes in angle (each additional 90° adds 2.5 distance units. (See figure.)

Ex8. The curve goes out (the distance increases) .2778 units of distance every step of 10 degrees, going 10 units of distance out from center to the outside edge during the 36 steps while going around the circle once. 36 steps x .2778 units of distance/step = 10. All the values given in the answer to Exercise 12. (See figure.)

Ex9. For rounding down, 10 units of distance / .25 units of distance/step = 40; 40 steps x 10°/step = 400°, or 40° more than the 360° to go around the circle once. At step 36 (360°), the distance from the center is 9 units. (See figure.)
For rounding up, 10/.33 = 30 steps, and so the curve diverges to the edge of the circle in 30 steps, going around 300°, just 30° past 3/4 of the way around the circle. If the spiral were continued out to 360°, the spiral would be 12 units of distance from the center (36 x .3333 = 12).
These constructions and calculations show how a small error can get bigger (diverge), as one goes around the curves or trajectories. (See figure showing divergence when two spirals are superimposed on the same graph.) Note that this concept, divergence, and this word trajectory, are new. We will be dealing with this concept more later on.

Ex10. Cut the distance to be added at each step in half (.13333...); a third (.0925925...). (See figure.)

Ex11.

Ex12. We had to compute these for Exercise 8.
r₄=1.111..., r₁₀=2.777..., r₁₈=5.000..., r₂₇=7.500..., r₃₆=10.000...,

The whole sequence to 3 decimal places is:

r_1-9=0.278, 0.556, 0.833, 1.111, 1,389, 1.667, 1.944, 2.222, 2.500
r_10-18=2.778, 3.056, 3.333, 3.611, 3.889, 4.167, 4.444, 4.722, 5.000
r_19-27=5.278, 5.556, 5,833, 6.111, 6.389, 6.667, 6.944, 7.222, 7.500
r_28-36=7.778, 8.056, 8.333, 8.611, 8.889, 9.167, 9.444, 9.772, 10.000

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Other Places To Go

Polar Coordinates

Making Spirals

Math of Spirals

Figures of Spirals

Defs Rel to Spirals

Spiral Index Page

Syn-Dyn Index Page

Created: 12/23/96 Updated: 1/11/97